It appears that the jQuery used on your site is a custom build that does not have a Date Picker included. Can you confirm that you have a custom build of jQuery 1.4.2 that has custom UI 1.8.2?

In can case, we have updated our module to work in your situation to Advanced Login 45.02.09 - That will not reference the date picker UI unless needed. This will only be a solution if your registration form does not have any dates.

The second solution is to update the jQuery to the current full release version. This will then allow the reference to the date picker to be active.

effectively, to minimize the size of jquery.ui compile a customized version with minimal
functionality to work correctly the module. Wait for the release of version 45.02.09.
Thank you very much for your help.